Labelled tree graphs, Feynman diagrams and disk integrals

Understand

In this note, we introduce and study a new class of "half integrands" in Cachazo-He-Yuan (CHY) formula, which naturally generalize the so-called Parke-Taylor factors; these are dubbed Cayley functions as each of them corresponds to a labelled tree graph.

• The CHY formula with a Cayley function squared gives a sum of Feynman diagrams, and we represent it by a combinatoric polytope whose vertices correspond to Feynman diagrams.
• We provide a simple graphic rule to derive the polytope from a labelled tree graph, and classify such polytopes ranging from the associahedron to the permutohedron.
• Furthermore, we study the linear space of such half integrands and find (1) a nice formula reducing any Cayley function to a sum of Parke-Taylor factors in the Kleiss-Kuijf basis (2) a set of Cayley functions as a new basis of the space; each element has the remarkable property that its CHY formula with a given Parke-Taylor factor gives either a single Feynman diagram or zero.